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Unitary representations correspond to nondegenerate star representations of L one

Statement

Assume the Axiom of Choice. Let $G$ be an LCH group. The assignment $π \mapsto σ_{π}$ , $σ_{π} (f) : = π (f)$ , from strongly continuous unitary representations of $G$ to star-representations of the Banach $*$ -algebra $L^{1} (G)$ (Strongly continuous unitary representations, invariant linear subspaces and intertwiners, The integrated form of a unitary representation, Nondegenerate star-representations of a Banach star-algebra) is a bijection, respecting unitary equivalence, between unitary representations of $G$ up to unitary equivalence and nondegenerate star-representations of $L^{1} (G)$ up to equivalence: every unitary representation has a nondegenerate integrated form, every nondegenerate star-representation is the integrated form of a unique unitary representation, and a unitary intertwines two representations if and only if it intertwines their integrated forms.

Facts & Assumptions

Given: AC; an LCH group $G$ ; strongly continuous unitary representations of $G$ ; nondegenerate star-representations of $L^{1} (G)$ .

[F1]

The integrated form $f \mapsto π (f)$ of a unitary representation is a contractive nondegenerate star-representation of $L^{1} (G)$ , and $π (f) ξ$ is characterised by $⟨ π (f) ξ, η ⟩ = \int_{G} f (g) ⟨ π (g) ξ, η ⟩ d g$ (Integrated forms are contractive nondegenerate star representations of L one, The integrated form of a unitary representation).

[F2]

Conversely, every nondegenerate star-representation $σ$ of $L^{1} (G)$ on a Hilbert space $K$ is the integrated form of a unique unitary representation $U$ of $G$ ; it satisfies $U (g) σ (f) ξ = σ (L_{g} f) ξ$ for all $g, f, ξ$ (Recovering a unitary group representation from a nondegenerate L one representation).

[F3]

$L^{1} (G)$ has a two-sided approximate unit $(e_{U})$ with $∥ e_{U} ∥_{1} \leq 1$ , $e_{U} * f \to f$ , $f * e_{U} \to f$ (L1 group algebras have a contractively bounded approximate identity).

[F4]

$L_{g} f (x) = f (g^{- 1} x)$ defines an isometric linear map of $L^{1} (G)$ with $L_{g} (a * b) = (L_{g} a) * b$ and $L_{g} L_{h} = L_{g h}$ (Recovering a unitary group representation from a nondegenerate L one representation); moreover $π (L_{g} e_{U}) ξ \to π (g) ξ$ for every unitary representation $π$ and every $ξ$ , because left invariance gives $π (L_{g} e_{U}) ξ = \int_{G} e_{U} (y) π (g y) ξ d y$ and $e_{U} \geq 0$ has total mass one and support shrinking to ${e}$ , so the norm difference is at most $\sup_{y \in U} ∥ π (g y) ξ - π (g) ξ ∥$ . [F1, F3]

Proof

technique · direct

Given: AC, an LCH group $G$ , and the integrated-form assignment $π \mapsto σ_{π}$ .

1.1F1F2F4

The assignment is well defined on equivalence classes and preserves equivalence: by [F1] each $σ_{π}$ is a nondegenerate star-representation; and if $U : H_{π} \to H_{ρ}$ is a unitary intertwiner, $U π (g) = ρ (g) U$ , then the defining weak integrals give $U σ_{π} (f) = σ_{ρ} (f) U$ for every $f \in L^{1} (G)$ , since $U$ passes through the integral. Conversely, if $U$ intertwines the integrated forms, it intertwines each $π (L_{g} e_{V})$ with $ρ (L_{g} e_{V})$ ; their strong limits are $π (g)$ and $ρ (g)$ by [F4], so $U π (g) = ρ (g) U$ . Thus a specified unitary intertwines the group representations exactly when it intertwines their integrated forms.

1.2F1F3F4

For every unitary representation $π$ , every $g \in G$ , $f \in L^{1} (G)$ and $ξ \in H_{π}$ : $π (g) π (f) ξ = π (L_{g} f) ξ$ . Indeed, $L_{g} (e_{U} * f) = (L_{g} e_{U}) * f$ by [F4], so $π (L_{g} (e_{U} * f)) = π (L_{g} e_{U}) π (f)$ by multiplicativity [F1]; as $L_{g} (e_{U} * f) \to L_{g} f$ in $L^{1} (G)$ and $π (L_{g} e_{U}) \to π (g)$ strongly by [F4], both sides converge to $π (L_{g} f) ξ$ and $π (g) π (f) ξ$ respectively.

1.3F2

Surjectivity: given a nondegenerate star-representation $σ$ , [F2] produces a unitary representation $U$ with $π_{U} (f) = σ (f)$ for all $f$ , so every nondegenerate star-representation is an integrated form.

2.1F2step 1.2

Injectivity: suppose unitary representations $π$ and $π^{'}$ have the same integrated form $σ$ . By [F2] applied to $σ$ there is a unique unitary representation $U$ with $U (g) σ (f) ξ = σ (L_{g} f) ξ$ ; by step 1.2 both $π$ and $π^{'}$ satisfy this identity, because $σ (f) = π (f) = π^{'} (f)$ ; hence $π = U = π^{'}$ . Thus the assignment is injective on equivalence classes.

3.1F1F2step 1.1step 1.3step 2.1∎

Combining steps 1.1, 1.3 and 2.1, the assignment induces a bijection between unitary equivalence classes of unitary representations and equivalence classes of nondegenerate star-representations, and step 1.1 shows exactly that it respects unitary equivalence in both directions. The Axiom of Choice is carried by the integrated-form correspondence of [F1]–[F2] as declared throughout this page (The Axiom of Choice).

Depends on

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Sources

Bachir Bekka and Pierre de la Harpe, Unitary Representations of Groups, Duals, and Characters (arXiv:1912.07262v1, 16 December 2019) (standard reference, not scraped)
Bachir Bekka, Pierre de la Harpe and Alain Valette, Kazhdan's Property (T) (Cambridge University Press 2008; author-hosted complete text) (standard reference, not scraped)